Step
*
2
1
1
1
1
1
of Lemma
bag-member-count
1. T : Type
2. eq : EqDecider(T)
3. x : T
4. T List ∈ Type
5. ∀as,b1:T List. (permutation(T;as;b1) ∈ Type)
6. ∀as:T List. permutation(T;as;as)
7. L : T List
8. b1 : T List
9. permutation(T;L;b1)
10. ((1 ≤ #([y∈L|eq x y]))
⇒ x ↓∈ L) = ((1 ≤ #([y∈b1|eq x y]))
⇒ x ↓∈ b1) ∈ Type
11. h : 1 ≤ ||filter(λy.(eq x y);L)||
12. uiff(↑null(filter(λy.(eq x y);L));∀[i:ℕ||L||]. (¬↑(eq x L[i])))
13. ∃i:ℕ||L||. (↑(eq x L[i]))
⊢ (x ∈ L)
BY
{ ((Unfold `l_member` 0 THEN ParallelLast) THEN Auto) }
1
1. T : Type
2. eq : EqDecider(T)
3. x : T
4. T List ∈ Type
5. ∀as,b1:T List. (permutation(T;as;b1) ∈ Type)
6. ∀as:T List. permutation(T;as;as)
7. L : T List
8. b1 : T List
9. permutation(T;L;b1)
10. ((1 ≤ #([y∈L|eq x y]))
⇒ x ↓∈ L) = ((1 ≤ #([y∈b1|eq x y]))
⇒ x ↓∈ b1) ∈ Type
11. h : 1 ≤ ||filter(λy.(eq x y);L)||
12. ∀[i:ℕ||L||]. (¬↑(eq x L[i])) supposing ↑null(filter(λy.(eq x y);L))
13. ↑null(filter(λy.(eq x y);L)) supposing ∀[i:ℕ||L||]. (¬↑(eq x L[i]))
14. i : ℕ||L||
15. ↑(eq x L[i])
16. i < ||L||
⊢ x = L[i] ∈ T
Latex:
Latex:
1. T : Type
2. eq : EqDecider(T)
3. x : T
4. T List \mmember{} Type
5. \mforall{}as,b1:T List. (permutation(T;as;b1) \mmember{} Type)
6. \mforall{}as:T List. permutation(T;as;as)
7. L : T List
8. b1 : T List
9. permutation(T;L;b1)
10. ((1 \mleq{} \#([y\mmember{}L|eq x y])) {}\mRightarrow{} x \mdownarrow{}\mmember{} L) = ((1 \mleq{} \#([y\mmember{}b1|eq x y])) {}\mRightarrow{} x \mdownarrow{}\mmember{} b1)
11. h : 1 \mleq{} ||filter(\mlambda{}y.(eq x y);L)||
12. uiff(\muparrow{}null(filter(\mlambda{}y.(eq x y);L));\mforall{}[i:\mBbbN{}||L||]. (\mneg{}\muparrow{}(eq x L[i])))
13. \mexists{}i:\mBbbN{}||L||. (\muparrow{}(eq x L[i]))
\mvdash{} (x \mmember{} L)
By
Latex:
((Unfold `l\_member` 0 THEN ParallelLast) THEN Auto)
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